Topic: Serious Discussion / Logic

Originally posted bytenco1:Originally posted byslogsdon:has anyone in here actually formally studied logic?

i’m guessing not

…

Wait, you can study logic?

You can study logic in 3 or so ways:

1) from the philosophy side, with questions like “what are correct ways of inferring information”, and “how do proof, truth, and reality all relate to each other”;

2) from the math side, with questions like “what system should we use to talk about and interpret this particualr mathematical structure”, “what are the properties of this system”, and “what can we learn about a structure once we know it can be interpreted in a certain system”;

3) from the algorithm side, with questions like “how can we efficiently implement the rules for a system on a computer” and “how difficult is it to solve problems like ____ for structures interpretable in systems like _____”

I’m sure there’s more, but this is probably enough to vaguely cover the work of most people who would describe themselves as a logician. All of these people would have learned formal logic, and at least for 2 and 3 routinely work with it.

I’m a card carrying member of category 2 (and occasionally sit around listening to category 3 people because those are also often math people). So I’ll ramble from that perspective.

Doing it from the math side means we’re already starting with assumed logic and counting ability on everyone’s part, and then you go and define formal systems called logics. The assumed ambient logic and number ideas isn’t like “you need to go learn that in order to learn this”, it’s something that we have to assume is just a general reality in order to even make sense doing math at all, sort of like how you have to assume English conveys any of your intended meaning to think that having a conversation has a point.

The general theme with what makes something “a logic” is that it gives you a formal language and it tells you what kinds of deduction are okay.

Crack open a math logic book and you’ll see a definition of “proof” somewhere in there. Usually it’s something like “proof in the formal system F” is defined as a finite list of formulas from F such that each formula is either an axiom or is derived from the earlier formulas using the deduction rules. So a proof is usually a thing that looks like

1) By assumption, blah blah blah. 2) By assumption, blah blah. 3) By 1 and 2, and some inference rule, blah blah blah blah blah. 4) By 3 and some inference rule, blah blah blah.

You can go find a definition of “true” in a math logic book too, but it would probably seem weird unless you learn enough stuff before.

Being “true” and being “provable” aren’t always related, though you’d want them to be for your logic to be useful. Showing this for the system that’s commonly used as sort of the “base logic” for most math reasoning isn’t totally obvious. For things like the theory of arithmetic, there’s weird stuff like we know some sentences are “true” but are not “provable”.

Anyway, a potentially interesting point to take away here is that this is all somewhat arbitrary because the definitions are up to people to make. So being “proved” and being “true” depend on the rules you decided to use. But we don’t just pick them totally randomly. We generally try to pick rules that match with that “obvious reasoning”, but there’s no absolute justification for that, and that’s where philosophy comes in.

When people talk about “logic” in everyday use, they probably mean that general system everyone is assumed to understand, knowing what “and” “or” “implies” “for all” “at least one” “not” etc. mean. But, when they say that they used logic or proved something or that something is true, it’s often more accurate to say this relative to *their* system, with whatever beliefs and assumptions they were using. If it’s transparent enough what they did and it overlaps with your system enough, you’d agree with them. Otherwise, you might not. This is sometimes why you get rational people disagreeing on moral issues for example. Though eventually they should figure out that they’re disagreeing about assumptions or acceptable methods of inference. Again, the question of whether one system is better than another is a topic for philosophy.